Recall: Factored Form: y=a(x-s)(x-t)
Standard Form: y=ax2 +bx+c
To convert the equation of a quadratic relation from factored form to standard form, we will expand and simplify the factored form.
y = (x + 3)(x + 2)
y = x2 + 2x + 3x + 6
y = x2 + 5x +6
Use FOIL - First Outside Inside Last
y = (x - 4)2 Expanding a perfect square.
y = (x - 4)(x - 4)
y = x2 - 4x - 4x + 16
y = x2 - 8x + 16
HW: p. 298 # 2ad, 3agj, 5ag, 7a, 8bd, 9a, 10ce
3.7 Common Factoring
The first step in factoring a trinomial is to check whether there is a common factor to all three terms.
eg. 4x2 - 12x + 8 we can factor out the 4, to get
4(x2 - 3x + 2)
which can then be factored
4(x - 2)(x-1)
3.8 Factoring Monic Quadratics
Monic Trinomials are trinomials where a = 1.
The two main methods are Guess and Check, and PSI (Product Sum Integer)
Guess and Check is just using FOIL in reverse.
PSI Method, is the method of finding two integers whose product is 'c' and sum is 'b'
e.g. x2 - 2x - 63 we need to find two integers whose product is -63 and sum is -2
these would be -9 and 7, these will be the numbers to use in our binomials.
(x - 9)(x + 7)
3.9 Factoring Complex Quadratics
Complex Quadratics is when a not equal to 1.
First check if common factor. Often an expression can look complex but can be turned into a monic when you remove the common factors.
Methods 1. Guess and check, if a and c are primes it is often easy just to try the possibilities.
2. Decomposition - this is a common method. See text book or Internet for further info.
3. Black Box method - this seems to be the preferred method, so will explain further.
Step 1. Find two integers who product is ac and sum is b. (same as decomposition, and similar to PSI)
Step 2. "Divide" these integers by "a", creating a "fraction"
Step 3. Simplify the fraction.
Step 4. We can now "read up" the answer, with the denominators being the coefficient of x, and the numerator being the constant in the equation).
Example 6x2 - 17x + 12
we need to find two integers whose product is 72 ( 6 x 12) and add up to -17
we would try the pairs 1, 72 then 2, 36, then 3, 24, then 4, 18, then 6, 12 until we reach 8 and 9.
Change the signs to negative since they have to add up to -17
So our numbers are -8 and -9
Step 2 "Divide" by 'a' -8/6 and -9/6
Step 3 Simplify -4/3 and -3/2
The answer is (3x - 4) (2x -3) [note the coefficients of x coming from the denominators]
Verify in your head by using FOIL and it works?
This method works very well. It is similar to decomposition.
3.6 Factored Form to Standard Form
Recall: Factored Form: y=a(x-s)(x-t)
Standard Form: y=ax2 +bx+c
To convert the equation of a quadratic relation from factored form to standard form, we will expand and simplify the factored form.
y = (x + 3)(x + 2)
y = x2 + 2x + 3x + 6
y = x2 + 5x +6
Use FOIL - First Outside Inside Last
y = (x - 4)2 Expanding a perfect square.
y = (x - 4)(x - 4)
y = x2 - 4x - 4x + 16
y = x2 - 8x + 16
HW: p. 298 # 2ad, 3agj, 5ag, 7a, 8bd, 9a, 10ce
3.7 Common Factoring
The first step in factoring a trinomial is to check whether there is a common factor to all three terms.
eg. 4x2 - 12x + 8 we can factor out the 4, to get
4(x2 - 3x + 2)
which can then be factored
4(x - 2)(x-1)
3.8 Factoring Monic Quadratics
Monic Trinomials are trinomials where a = 1.
The two main methods are Guess and Check, and PSI (Product Sum Integer)
Guess and Check is just using FOIL in reverse.
PSI Method, is the method of finding two integers whose product is 'c' and sum is 'b'
e.g. x2 - 2x - 63 we need to find two integers whose product is -63 and sum is -2
these would be -9 and 7, these will be the numbers to use in our binomials.
(x - 9)(x + 7)
3.9 Factoring Complex Quadratics
Complex Quadratics is when a not equal to 1.First check if common factor. Often an expression can look complex but can be turned into a monic when you remove the common factors.
Methods 1. Guess and check, if a and c are primes it is often easy just to try the possibilities.
2. Decomposition - this is a common method. See text book or Internet for further info.
3. Black Box method - this seems to be the preferred method, so will explain further.
Step 1. Find two integers who product is ac and sum is b. (same as decomposition, and similar to PSI)
Step 2. "Divide" these integers by "a", creating a "fraction"
Step 3. Simplify the fraction.
Step 4. We can now "read up" the answer, with the denominators being the coefficient of x, and the numerator being the constant in the equation).
Example 6x2 - 17x + 12
we need to find two integers whose product is 72 ( 6 x 12) and add up to -17
we would try the pairs 1, 72 then 2, 36, then 3, 24, then 4, 18, then 6, 12 until we reach 8 and 9.
Change the signs to negative since they have to add up to -17
So our numbers are -8 and -9
Step 2 "Divide" by 'a' -8/6 and -9/6
Step 3 Simplify -4/3 and -3/2
The answer is (3x - 4) (2x -3) [note the coefficients of x coming from the denominators]
Verify in your head by using FOIL and it works?
This method works very well. It is similar to decomposition.
3.10 Summary of Factoring